December 14, 2012

Fig. 1: Transmission Line's lumped circuit
equivalent. |

We start to care about the transmission line behavior of wires when the rise and fall times of the logic signals get short compared to the length of the wires or the transmission lines on which they travel. Two of the major adverse effects are reflections and crosstalk. Lets talk about the problems a little more to set up the motivation for the analysis of the transmission lines in a later section.

The electromagnetic energy launched down a transmission line as a signal must be absorbed as it arrives at the end of the transmission line or it will be reflected back to the source resulting in overshoot that could result in the voltage to exceed the maximum input voltage rating of the input. [1] Since most signal lines terminate in logic inputs at their receive ends, they are effectively open circuit. Under these conditions the EM wave is reflected back to the source without getting inverted. This is seen as doubling of the signal at the receiver that can result in a circuit failure by overstressing the insulating oxides or causing a parasitic junction to conduct. Even a clock speed of only 1 Hertz can result in malfunctions. The fastest rise times of the components used in a design determine what level of discipline is needed in managing transmission lines, and not the clock frequency being considered. The second problem that is usually encountered when speeds increase is crosstalk between signals that run parallel to each other. Backward crosstalk increases linearly with the length that two transmission lines run parallel up to a point where saturation is reached. Beyond this length, known as the critical length, continuing to run in parallel results in no additional backward crosstalk. The forward crosstalk depends on the rate of change of the input signal and the polarity of the forward crosstalk signal depends on the values of the capacitive and inductive coupling coefficients.

From the electromagnetic (EM) theory, in order to achieve high efficient point-to-point transmission of power and information, the source energy must be guided. [2] Wires used to transmit high frequency communication signals, can be regarded as transmission lines which guide the Transverse Electromagnetic (TEM) waves along them. The two intrinsic line parameters, the characteristic impedance and the propagation constant, are derived based on the transmission line theory. According to Chang, the two-wire transmission line must be a pair of parallel conducting wires separated by a uniform distance. [3] The paired line traces are regarded as a distributed parameter network, where voltages and currents can vary in magnitude and phase over its length. Hence, it can be described by circuit parameters that are distributed over its length.

In Fig. 1, the quantities v(z, t) and v(z+Δz, t) denote the instantaneous voltages at location z and z+δz respectively. Similarly, i(z, t) and i(z+Δz, t) denote the instantaneous currents at z and z + Δz , respectively. R defines the resistance per unit length for both conductors (in Ohm/m); L defines the inductance per unit length for both conductors (in H/m); G is the conductance per unit length (in mho/m); C is the capacitance per unit length (in F/m). Applying the Kirchhoff s voltage law and current law respectively, the following two equations can be obtained,

v(z,t) - R Δz i(z,t) - L Δz ∂i(z,t)/∂t - v(z+Δz,t) = 0 | (1) |

i(z,t) - G Δz v(z+Δz,t) - C Δz ∂v(z+Δz,t)/∂t - i(z+Δz,t) = 0 | (2) |

If i and v are expressed in the phasor form, i.e.
v(z,t) = Re[V(z)e^{jωt}] and i(z,t) =
Re[I(z)e^{jωt}], and when Δz→0, the time
harmonic line equations can be derived from (1) and (2),

- dV(z)/dz = (R+jωL) I(z) | (3) |

- dI(z)/dz = (G+jωC) V(z) | (4) |

Combine (3) and (4), the line voltage and current in terms of position z can be expressed in the following differential equations:

d^{2}V(z)/dz^{2} =
γ^{2} V(z) |
(5) |

d^{2}I(z)/dz^{2} =
γ^{2} I(z) |
(6) |

where,

γ=
α + j β =
[(R + jωL)(G + jωC)]^{1/2} |
(7) |

γ is the propagation constant whose real and imaginary parts, α and β, are the attenuation constant and phase constant (in rad/m) respectively. The time harmonic V(z) and I(z) can be expressed in another form:

V(z) = V^{+}(z) +
V^{-}(z) =
V_{0}^{+} e^{-γz} +
V_{0}^{-} e^{γz
} | (8) |

I(z) = I^{+}(z) +
I^{-}(z) =
I_{0}^{+} e^{-γz} +
I_{0}^{-} e^{γz
} | (9) |

where the plus and minus superscripts denote waves
traveling in the +z and -z directions, respectively. Wave amplitudes
(V_{0}^{+}, I_{0}^{+}) and
(V_{0}^{-}, I_{0}^{-}) are related by
(3) and (4). We have
(V_{0}^{+})/(I_{0}^{+}) = -
(V_{0}^{-})/(I_{0}^{-}) = (R +
jωL)/γ. For an infinite line (actually a semi-infinite line
with the source at the left hand side end) the ratio of the voltage and
current at any z for an infinite long line, V(z)/I(z) =
V^{+}(z)/I^{+}(z) =
V_{0}^{+}/I_{0}^{+}, gives the
characteristic impedance of the line,

Z_{0} =
[(R + jωL)/(G + jωC)]^{1/2} |
(10) |

Note that the propagation constant y and
characteristic impedance Z_{0} are characteristic properties of
a transmission line whether or not the line is infinitely long. They
depend on R, L, G, C and ω but not the length of the line.

With this model, system designers and interested people can get a simple understanding of the transmission line behavior, their impact and the first pass analysis and appreciate the importance of not only the components and circuit design which makes a PCB but the routing as well. The input impedance plays an important role in designing the transceiver matching circuits and determining the transducer gain and signal power.

© Shiv Agarwal. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] L. W. Ritchey, *Right the First Time*
(Speeding Edge, 2003)..

[2] H. Meng *et al.*,
"A
Transmission Line Model for High-Frequency Power Line Communication
Channel," Proc. Intl. Conf. Power Systems Technology, PowerCon 2002,
**2**, 1290 (2002).

[3] D. K. Cheng, *Fundamental of Engineering
Electromagnetics* (Prentice Hall, 1992).